Radiology

ABSTRACT

In a method of constructing a representation of the distribution of absorption in a planar region of a body interpolation is provided, between data signals representing absorption of radiation along parallel paths, to provide signals for an increased number of such paths as required for the processing used. The interpolation is performed by forming differences of second or higher order, subdividing the differences and then combining them by running additions.

This is a continuation of appln. Ser. No. 596,623, filed July 17, 1975,now U.S. Pat. No. 4,002,910 issued July 11, 1977.

The present invention relates to a method of constructing arepresentation of the distribution of absorption with position in aplanar region of a body.

In U.S. Pat. No. 3,778,614 there is described apparatus of that typeincluding a source of penetrating radiation and detector meansresponsive to the radiation. The source and detector means are scanned,in the plane of the slice and relative to the body, so that the detectormeans provides output signals indicative of the absorption suffered bythe radiation as it traverses many different paths through the body inthe said plane. The radiation beam paths are grouped in sets of parallelpaths, such as that indicated in FIG. 1 by dashed lines representing thecentrelines of the paths, each set being at a different angle relativeto the body. The paths of each set do not need to be strictly parallelbut should not deviate too much from parallelism. In the said UnitedStates Patent the data signals relating to each of the beam paths areprocessed by a method of successive approximations. The finalrepresentation is thus built up in a final so-called matrix store inaddresses corresponding to the centre points of meshes of a notionalcartesion meshwork, regarded as lying in the said planar region. Such ameshwork has also been shown in FIG. 1 superimposed on the illustrativeparallel set of beam paths. It should be understood, however, that manymore beam paths and mesh elements would be used in practise.

An alternative form of processing is described in U.S. Pat. No.3,924,129. In that method, which is of a convolution character, the dataappropriate to the beam paths of a parallel set are first processed toallow the image to be formed in a so-called layergram manner. After suchconvolution an accurate image is formed by summing in each store addressthe convolved data values, f₁, f₂ etc. for all beam paths, one from eachparallel set, whose centrelines pass through or close to the appropriatemesh centre point. Unfortunately, although the beam paths may all passthrough mesh centre points for one orientation of the scan, that is notgenerally true for all scanning angles as can be seen from FIG. 1. Theset shown, inclined at angle θ to the vertical of the mesh, comprisesten beams, spaced along the axis of a set parameter r in a directionperpendicular to the beams. The appropriate convolved data values areshown as f₁ to f₁₀. It can be seen that the third beam from the left,associated with datum f.sub. 3, passes through a centre point at a sothat the datum f₃ may be applied to the address for that mesh. Howeverthe beam misses a centre point at b by a distance x which is less thanthe beam centreline spacing so that there is no datum for mesh b.

To provide data for each such mesh it is necessary to interpolatebetween the available data to provide, for example, thirty interpolatedbeam absorption values between each derived pair such as f₃ and f₄. Thusone interpolated or derived beam path will pass through each mesh centrepoint or sufficiently close for the absorption value to be allocated tothat point without significant error.

As a result partly of the distribution of radiation imposed on the beamsby collimator means, employed to define the beams, and partly of thecontinuous motion of the beam during scanning, each beam tends to have asinusoidal distribution of intensity. This may extend skirt-to-skirtover a range equal to twice the spacing of the beam paths, i.e. twicethe sampling interval, as shown in FIG. 1 for some of the pathsillustrated. It may, however, in some circumstances extend over two ormore of the adjacent beam paths.

Interpolation has been performed in terms of that effective sinusoidaldistribution of the contributing beams. In that case, a proportion ofthe absorption suffered by one of these two beams contributes to themesh point absorption value according to a sinusoidal function of thedisplacement of the beam from that point. A complementary proportion ofthe absorption suffered by the other of the two beams also contributesto this value. This type of interpolation is accurate in the sense thata uniform field of original absorption yields on interpolation, as itshould, a uniform pattern of absorption in the final reconstruction. Themethod however fully reflects statistical errors of data measurementinto this reconstruction. This is in contrast with the fact that at thefrequency of such measurement, namely the sampling rate, the response ofthe system as a whole, following recognised sampling principles, is zeroin regard to true information.

It is an object of the present invention to provide an improved methodof reconstruction with a reduced response to such statistical errors.

It is another object of the invention to provide a method ofreconstruction capable of relatively more rapid operation.

According to the present invention there is provided a method ofconstructing a representation of the distribution of absorption in aplanar region of a body including the steps of providing data signalsrepresenting the absorption of the radiation along a plurality ofsubstantially parallel paths in the plane of the said region,interpolating between the said data signals to provide further datasignals representing the absorption of the radiation along an increasednumber of such paths and allocating, to each of a plurality ofpredetermined points in a field notionally delineated in the saidregion, the data signal or further data signal associated with the pathwhose centreline passes closest to that point, wherein the interpolationis achieved by forming signals representing differences of said dataextending to order m, where m is at least two, subdividing some of saiddifference signals and combining the subdivided differences by runningaddition.

In order that the invention may be better understood it will now bedescribed by way of example with reference to the accompanying drawingsin which:

FIG. 1 has the significance explained hereinbefore,

FIG. 2 shows a typical X-ray apparatus as referred to hereinbefore.

FIG. 3 shows in block diagrammatic form a circuit in accordance with theinvention; and

FIG. 4 and 5 are graphs showing the character of the interpolationaccording to the invention.

Referring now to FIG. 2 there is shown in end elevation an X-rayapparatus of the type described and in conjunction with which theinvention is employed. A turntable 1, having a central aperture 2, inwhich a body 3 to be examined is inserted, is arranged so as to becapable of rotation about an axis 4 perpendicular to its plane andcentrally disposed in the aperture. The turntable is rotated by a motor5 which drives a gear wheel 6a, the wheel 6a being arranged to engagecogs formed around the periphery 7 of the turntable 1. The gear wheel 6ais journalled in a fixed main frame of the apparatus, not shown, and theturntable is further supported by two undriven gear wheels 6b and 6calso journalled in the main frame.

The body 3 remains fixed while the turntable 1 rotates about it. A twopart ring member 8a and 8b is fitted around the region of the body 3 tobe examined and the ring is secured to a patient supporting structurewhich comprises a two part bed, one part on either side of the turntableso as to hold the patient securely in position such that the exploringradiation can traverse the region of interest. Only one part of thesupporting structure 9 is shown in the drawing for the sake of clarity.Disposed around the patient in the region of interest and trapped inposition by the ring 8a, 8b is a material 10, for example water in abag, which absorbs the radiation to an extent similar to body tissue.The material 10 helps to exclude air from the region around the body andalso assists in the accommodation of patients of differing sizes, sincethe apparatus treats the entire content of the ring 8a, 8b as being abody.

The turntable 1 carries two compensating members 11 and 12 fixedthereto. These members are arranged to provide a substantially uniformabsorption to the radiation for all beam paths of a lateral scan despitethe circular cross section of the "body" of ring 8a, 8b. Thus it isensured that any variations of absorption are caused substantially onlyby variations in the body 3.

Also fixedly secured to the turntable 1 is a reversible motor 11a whichdrives a toothed belt 12a by means of a drive shaft 13 journalled inturntable 1. The belt 12a also passes over an idler wheel 14 alsojournalled in turntable 1. Secured to the belt 12a is a source 15 of asingle beam of radiation 16, in this example, although it may also be afan shaped swath as described in U.S. Pat. No. 3,946,234. The source isdriven to and fro laterally by belt 12a, being mounted on a bearingtravelling on a track 17. A counter balance weight 18 is fixed to theopposite side of belt 12a to compensate for out of balance forces duringthe lateral movement.

Linked to the source 15 by a light weight but rigid yoke 19 is acollimator/detector unit 20 which may contain, for example, ascintillator crystal and photomultiplier. In the event that the source15 provides a fan shaped swath of radiation, the detector unit 20comprises a plurality of such crystals and photomultipliers, defining aplurality of beam paths in the fan, as described in the aforesaidapplication Ser. No. 502,080. The detector unit 20 also moves on abearing on a track 21 on turntable 1. Yoke 19 carries a graticule 22which cooperates with a light source and photocell 23 to provide signalsindicative of the progress of the lateral scan. The graticule 22 is atranslucent strip carrying engraved lines which interrupt the light pathbetween the light source and the photocell. The signals obtained areused by a computer controlling the processing to determine the positionof the beam 16 in relation to the body 3 for each data value obtained bydetector 20. A similar graticule 24 fixed to turntable 1 and combinedwith a light source and photocell unit 25, fixed in a manner not shownto the main frame of the apparatus, indicates the progress of therotational movement for the same purpose. The apparatus is arranged tomake a lateral traverse to provide one parallel set of data as shown inFIG. 1 and to then make an orbital movement so as to repeat the lateralmovement at a different inclination. This process is repeated over, say,180° of rotation to build up the desired member of parallel sets.

The signals from detector unit 20 are amplified by an amplifier 26 andintegrated in an integrator 27 over a period determined by successivepulses from photocell 23. After further processing byanalogue-to-digital converter 28 and log converter 29 they aretransferred at 30 to a computer for processing as described in the saidU.S. Pat. No. 3,924,129. Units 26 to 29 which operate as described inthe said U.S. Pat. No. 3,924,129 or in the aforementioned U.S. Pat. No.3,778,614 may be said to comprise an initial processer indicated byreference numeral 32.

It will be understood that the invention is not restricted to use withan X-ray apparatus of the type described but may be used with any otherapparatus providing data which requires such interpolation. Other suchapparatus are those described in U.S. Pat. 3,937,963 or U.S. applicationSer. No. 544,799 filed Jan. 28, 1975, in which the parallel sets of dataare assembled from data received from a fan shaped spread of beam notsubject to a traverse scanning movement.

FIG. 3 shows, in block schematic form, a circuit for carrying out imagereconstruction. Reference 31 signifies the X-ray apparatus described inrelation to FIG. 2 and reference 32 is the initial processor alsodescribed. Absorption data derived from the scanner 31 are transferredvia the initial processor 32 to a data store 33 for storage in the formof parallel sets.

For the purpose of construction of the representation of linearabsorption for the region examined, the data signals are withdrawn,parallel set by parallel set, from store 33 and processed by aconvolution processor 34. Forms which this processor may take areindicated and described in detail in the specifications of U.S. Pat. No.3,924,129 and U.S. application Ser. No. 544,796 to achieve the resultdescribed hereinbefore. As each parallel set is processed theso-transformed data signals are stored in a manner also described thesaid specifications, in a storage means shown at 35 in FIG. 3.

Following the invention a differencing circuit 36 withdraws the storeddata held in store 35 for each parallel set in turn, and forms first andhigher order differences of the data values of the set, transferringthem to a difference store 37. The details of this differencing will bemore fully explained subsequently. In a manner also to be explained morefully, the difference values stored in store 37 are withdrawn for eachparallel set in turn by an integrator circuit 38. This circuit performsa functional integration in step-by-step manner in the intervals betweendifferenced values so as to derive absorption values appropriate to bestored at the addresses of matrix store 39. The nature of this store thefunction of which was described hereinbefore is also set out more fullyin the said U.S. application Ser. No. 544,796. As will become apparentthe integration process tends to exercise a smoothing effect upon thefunctional values distributed to the matrix store 39, so improving thediscrimination of the apparatus against statistical fluctuation in theoriginal acquisition of the data. The circuits 33 to 39 are representedin block form since in practice are provided by a digital computerincluding storage facilities, the computer being programmed to carry outthe functional operations about to be described in more detail to theextent that they have not already been indicated.

Referring to the action of the differencing circuit 6, as describedhereinbefore the convolved parallel set values relative to a particularset, as stored in store 5, are represented in order as

f₁, f₂, f₃, . . . , f₄, . . . , f_(N) at increasing values of r.

The kth first order difference of this series, represented as Δ_(k) ¹ isdefined by

    Δ.sub.k.sup.1 = f.sub.k+1 -f.sub.k

The circuit 3 forms all the differences of the series

    Δ.sub.1.sup.1, Δ.sub.2.sup.1, Δ.sub.3.sup.1, . . . , Δ.sub.k.sup.1, . . . , Δ.sub.N-1.sup.1

the kth difference of the second order, namely Δ_(k) ², is defined by

    Δ.sub.k.sup.2 = Δ.sub.k+1.sup.1 -Δ.sub.k.sup.1

and the circuit 3 also forms all the differences of the series

    Δ.sub.1.sup.2,Δ.sub.2.sup.2,Δ.sub.3.sup.2, . . . ,Δ.sub.k.sup.2, . . . ,Δ.sub.N-2.sup.2.

all these differences and, if desired, similar defined differences ofhigher order are stored, as they are formed, in the difference store 37in locations determined by the programme.

The character of the integrative interpolation performed by theintegrator circuit 38 on such differences will now be discussed for asimple example. Thus suppose the transformed data of a parallel set aswithdrawn from the store 35 to consist solely of five values f₁, f₂, f₃,f₄, f₅, equal respectively to the magnitudes

0,0,1,0,0.

These values relate to equally spaced values, of the set parameter r.

According to the expressions given hereinbefore it will be seen that thefirst order differences are given by

Δ₁ ¹ = 0,

Δ₂ ¹ = 1,

Δ₃ ¹ = -1 and

Δ₄ ¹ = 0,

                  TABLE 1                                                         ______________________________________                                                     Orig-   Orig-         Recon-                                                  inal    inal          structed                                         Orig-  first   second                                                                              Subdivided                                                                            first  Recon-                              Set   inal   order   order second  order  structed                            Param-                                                                              func-  differ- differ-                                                                             order   differ-                                                                              function                            eter  tion   ence    ence  difference                                                                            ence   values                              ______________________________________                                        r     f      Δ.sub.k.sup.1                                                                   Δ.sub.k.sup.2                                                                 Δ.sup.2.sub.k /n.sup.2 =Δ.sub.p.sup                               .2*     Δ.sub.p.sup.1*                                                                 φ(r)                            ______________________________________                                        0     f.sub.1 =0     0     0                                                  0.1                                                                           0.2                        0                                                  0.3                                                                           0.4                        0                                                  h/2=                                                                          0.5          0                     0                                          0.6                        1/25           0                                   0.7                                1/25                                       0.8                        1/25           1/25                                0.9                                2/25                                       h=                                                                            1.0   f.sub.2 =0     1     1/25           3/25                                1.1                                3/25                                       1.2                        1/25           6/25                                1.3                                4/25                                       1.4                        1/25           10/25                               3h/2=                                                                         1.5          1                     5/25                                       1.6                        -2/25          15/25                               1.7                                3/25                                       1.8                        -2/25          18/25                               1.9                                1/25                                       2h=                                                                           2.0   f.sub.3 =1     -2    -2/25          19/25                               2.1                                -1/25                                      2.2                        -2/25          18/25                               2.3                                -3/25                                      2.4                        -2/25          15/25                               5h/2=                                                                         2.5          -1                    -5/25                                      2.6                        1/25           10/25                               2.7                                -4/25                                      2.8                        1/25           6/25                                2.9                                -3/25                                      3h=                                                                           3.0   f.sub.4 =0     1     1/25           3/25                                3.1                                -2/25                                      3.2                        1/25           1/25                                3.3                                -1/25                                      3.4                        1/25           0                                   7h/2=                                                                         3.5          0                     0                                          3.6                        0                                                  3.7                                                                           3.8                        0                                                  3.9                                                                           4h=                                                                           4.0   f.sub.5 =0     0     0                                                  ______________________________________                                    

and the second order differences are given by

Δ₁ ² = 1

Δ₂ ² = -2 and

Δ₃ ² = 1

As mentioned hereinbefore higher order differences may be similarlyderived and interpolated values may be formed from many combinations ofsuch differences other than first order differences only. In thisexample, however, formation of interpolated values from only the secondorder differences will be described according to the preferredembodiment of the invention.

The initial function is rebuilt from intermediate values of r from thesecond order differences alone; Therefore, in order to provide thenecessary interpolated values, these differences are subdivided into nequal values, where 1/n is the value of the spacing of the increasednumber of function values to the signal number of such values. Sincesecond order differences are used, each subdivided difference is 1/n² ofthe corresponding original difference if the reconstructed function isto be correct. In a practical example n may have a value of thirty orforty depending on the number of interpolated values required. Howeverin this simplified example n will be given the value five. Thereconstruction of the function is thus as shown in Table 1 in which thedifferences have been shown in relation to values of the set variable r.The significance of the parameter h will be explained hereinbefore.

It will be seen that a more closely spaced set of second orderdifferences has been provided, there being in this example five for eachspan of the set parameter r associated with one of the original secondorder differences. The subdivided differences have been indicated byΔ_(p) ^(2*) where each value of Δ_(p) ^(2*) is equal to a value of Δ_(k)² /n². It should be understood that p indicates p^(th) difference in theset from an initial value and does not directly relate to a value of r.The value of Δ_(p=1) ^(2*) could be taken to be the zero value at r=0,however it is more convenient to set Δ_(p=1) ^(2*) at the first non-zerovalue, at r=0.6 in this example.

Using the values of Δ_(p) ^(2*) a more closely spaced set of first orderdifferences Δ_(p) ^(1*) are constructed by the reverse of the relationgiven above i.e.

    Δ.sub.p+1.sup.1* =Δ.sub.p.sup.1* +Δ.sub.p.sup.2*

The construction of this set of Δ_(p) ^(1*) is achieved by a runningaddition starting from an initial value of Δ_(p=1) ^(1*) which is zeroat a known position, namely that of r=0.5 in this example.

Similarly the new first order differences are used to construct a newmore closely spaced set of function values φ(r) by a similar runningaddition from a zero starting value at the value of r associated withΔ_(p=1) ^(2*). It will be seen that the newly constructed functioncomprises an increased number of more closely spaced valuescorrespondingly running smoothly among the original values, though notnecessarily passing through them.

Comparing Table 1 with FIG. 1 it can be seen that a value of thefunction for mesh point b between values f₃ and f₄ may now be moreaccurately obtained. Assuming x to be 0.4, on the scale of r given, therespective value would be φ.sub.(2.4) =3/5 for the 0,0,1,0,0, functionof the table.

The reconstruction can also be based on a series of running additionsbased on third or higher order differences. In general reconstructionfrom m'th order differences requires m such recurrent additions. Howevera single recurrent addition from first order differences will give adiscontinuous reconstructed function.

As mentioned hereinbefore, the integration may be based on such runningaddition using combinations of such differences, including the firstorder differences. A further example will be given to illustrate thatalternative approach and furthermore, to more properly indicate thenature of the process involved, the relationships used will be explainedin terms of finely divided steps approximating to continuousintegration. The function generated by the integration will berepresented as before by φ(r), and the equal spacing between originalfunction valves f will be denoted by h.

In the range of r lying between 1/2h and 3/2h the function φ(r), denotedin this range by φ₁ (r), is defined according to ##EQU1## Where u is thedifference between a value of the set variable r and the least value ofr in the range and t and t' are similarly variables on the scale of r.

In the range 3/2h to 5/2h the definition is correspondingly ##EQU2## andin the range 5/2h to 7/2h the definition in like character is ##EQU3##It should be understood that these expressions have been arbitrarilyderived to achieve the desired interpolation. Many similar expressionsmay be derived according to the general principles of the invention,namely to achieve a smooth interpolation by recurrent addition.

In these expressions the first and second order differences will havethe values given above if the function is the 0,0,1,0,0 function as usedin the previous example.

It follows in that case from the three expressions given that ##EQU4##and substituting its appropriate values of h and u ##EQU5##

In the ranges 0 to 1/2h and 7/2h to 4h the value of φ(r) is zero. FIG. 4gives the plot of the function over the total range 0 to 4h. It will beseen that the function represents a smooth curve which apart from theinitial and final values of the assumed data does not pass through thedata points but pursues a path lying between them. Thus theinterpolation technique of the invention tends to smooth the basic data,and to this extent it reduces the response of the system to statisticalfluctuations in the data, which the sinusoidal technique referred todoes not. On the other hand the technique of the invention allows a morecomplete response at the higher frequencies to true image informationthan does the sinusoidal method. It will be observed that the twoexamples given for the same initial function give similar results.

In practice, as stated, the integrator 38 operates on a basis ofrecurrent addition, rather than continuous integration, so that theintegrals in the second example given are formed as running summationsin finite steps. Correspondingly the differential dt has to be replacedby the finite difference Δt, which is chosen so that

    n. Δt = h

in which n is taken to be an integer as before. First order integralssuch as ##EQU6## thus become replaced by single summations of thecharacter ##EQU7## and in the same kind of way the second order integral##EQU8## becomes the double summation ##EQU9## In general an integral ofthe kth order becomes replaced by a summation of the same order in whichthe quantity summed is a kth order difference divided by n^(k). Theintegrator circuit 38 has thus first of all to perform the appropriatedivision with respect to n as in the first example. The similaritybetween this approach and that of the first example will be apparent butit must be stressed that they represent different and alternativemethods according to the invention.

The function derived above and plotted in FIG. 4 and the functionderived in the first example are both examples of interpolated functionsof an original 0,0,1,0,0 functions, which may be called a `stab`function. It will be apparent that any more complex function composed ofdiscrete values may be considered as a sum of a number of stab functionsof respective different amplitudes. Correspondingly the interpolation ofa more complex function may be achieved by the combination of aplurality of functions such as FIG. 4, each corresponding to one ofthose stab functions and with an appropriate peak amplitude.

In practice, however, any more complex function is interpolated directlyby a running summation as described above and not as a combination ofstab functions. As a further example a function having values f₁, f₂,f₃, f₄, and f₅ of 0,0,1,2,2, respectively will be interpolated in n = 10steps per span by the method of the second example using the threeexpressions given. The interpolation function φ(r) thus obtained isshown plotted in FIG. 5. In the figure the plot is shown forillustration as a continuous curve through the derived values but asused the function has the calculated values given in the last column ofthe table. Accordingly the value corresponding to a value of r mostappropriate to a matrix store address is the value used in theinterpolation for that address. It will be seen that although passingthrough the first and last of the given set values, these values beingdenoted by X's in the figure, the curve of the interpolation does notnecessarily pass through the remaining values, but assumes a smooth formthat at least lies close to them.

                                      TABLE 2                                     __________________________________________________________________________     r     f.sub.k                                                                         Δ.sup.1.sub.k                                                              ##STR1##                                                                         ##STR2##                                                                          ##STR3##                                                                         Δ.sup.2.sub.k                                                              ##STR4##                                                                            ##STR5##                                                                            ##STR6##                                                                            φ(r)                           __________________________________________________________________________    0     0                                   0                                     0.1      0                              0                                     0.2      0                              0                                     0.3      0                              0                                     0.4      0                              0                                     0.5   0  0                              0                                     0.6      0            .005              0.005                                 0.7      0            .005              0.020                                 0.8      0            .005              0.045                                 0.9      0            .005              0.080                               1     0    0         1  .005              0.125                                 1.1      0            .005              0.180                                 1.2      0            .005              0.245                                 1.3      0            .005              0.320                                 1.4      0            .005              0.405                                 1.5   1  0            .005              0.500                                 1.6         0.1             0           0.6                                   1.7         0.1             0           0.7                                   1.8         0.1             0           0.8                                   1.9         0.1             0           0.9                                 2     1       0.1    0        0           1.0                                   2.1         0.1             0           1.1                                   2.2         0.1             0           1.2                                   2.3         0.1             0           1.3                                   2.4         0.1             0           1.4                                   2.5   1     0.1             0           1.5                                   2.6         0.1                   -.005 1.595                                 2.7         0.1                   -.005 1.680                                 2.8         0.1                   -.005 1.755                                 2.9         0.1                   -.005 1.820                               3     2       0.1    -1             -.005 1.875                                 3.1         0.1                   -.005 1.920                                 3.2         0.1                   -.005 1.955                                 3.3         0.1                   -.005 1.985                                 3.4         0.1                   -.005 1.995                                 3.5   0     0.1                   -.005 2.000                                 3.6             0                       2                                     3.7             0                       2                                     3.8             0                       2                                     3.9             0                       2                                   4     2           0                       2                                   __________________________________________________________________________

In Table 2 the final column, giving values of the interpolation functionφ(r), is formed according to a technique which may be illustrated withreference to the range r=0.5 to r=1.5 corresponding to the range u=0 tou=1. For convenience ##EQU10## with n having the value 10, so that infact α has the value 0.01. It is evident that ##EQU11## and thatcorrespondingly

    S.sub.u+1 = 1/2(u.sup.2 + 2u+1)α= S.sub.u +uα+1/2α.

It follows that the first order difference

    Δ.sup.1 S.sub.u = u α + 1/2α,

and that the second order difference

    Δ.sup.2 S.sub.u = α.

More particularly it follows that ##EQU12## This recurrence formulatogether with the recurrence formula

    S.sub.u+1 = S.sub.u + Δ.sup.1 S.sub.u

is used to construct the final column of Table 2 on the basis of runningsummation.

At the outset, that is to say for u=0, it is given that

    S.sub.0 = 0,

    Δ.sup.1 s.sub.0 = 1/2 α,

    Δ.sup.2 s.sub.0 = α.

the final column of Table 2 thus builds up in accordance with thedetails of Table 3.

                  TABLE 3                                                         ______________________________________                                        u     Δ.sup.2 S.sub.u                                                                  Δ.sup.1 S.sub.u                                                                        S.sub.u                                         ______________________________________                                        0     α                                                                                 ##STR7##      0                                               1     α                                                                                 ##STR8##                                                                                     ##STR9##                                       2     α                                                                                 ##STR10##                                                                                    ##STR11##                                      3     α                                                                                 ##STR12##                                                                                    ##STR13##                                      4     α                                                                                 ##STR14##                                                                                    ##STR15##                                      ______________________________________                                    

it is to be noted that the value of α having been obtained by divisionwith respect to n in the first place, as mentioned, the value 1/2 αappearing in the top of the third column of Table 2 can be derivedsimply by the shift of one digit place in the register, if α is inbinary digital form. Thereafter the third column is built solely by arunning summation, and a running summation sufficies to construct thewhole of the fourth column giving the values of S_(u). The step-by-stepintegration of the invention is thus accomplished in these terms inrelation to Table 2 by utilising two accumulators.

The procedure given above using first and second order differences maybe adapted to avoid the one-digit shift at the commencement of Table 3so as to form the value 1/2 α. The adapted procedure, which is set outin Table 4 works entirely in multiples of α, while yielding aninterpolation function differing but very slightly from that calculatedin accordance with Table 3. In Table 4 the interpolation function, givenin the column headed S, is formulated at mid-points of the subspans in amanner similar to the first example above. The values of Δ¹ S_(u) inTable 3 are replaced by values decreased by the extent 1/2 α, as shownin the column of Table 4 under the heading Δ¹ S. Second order differencevalues are taken as before, as shown in the column designated Δ² S inTable 4, although positioned in the column as mid-subspan values. Theprocessing apart from the simplification that there is no initialone-digit shift is otherwise the same.

                  TABLE 4                                                         ______________________________________                                        u     Δ.sup.2 S                                                                         Δ.sup.1 S                                                                             S                                               ______________________________________                                        0               0                                                                   α                 0                                               1               0 + α = α                                               α                 0 + α = α                           2               α + α = 2 α                                       α                 α + 2α = 3 α                  3               2α + α = 3α                                       α                 3α  + 3α  = 6α                4               3α  + α = 4α                                ______________________________________                                    

The manner of interpolation used to illustrate the invention may betermed quadratic, since the highest order component of variation presentin the interpolation function in such illustration is that due to secondorder integration, namely that corresponding to the continuousintegration form ##EQU13## A greater measure of accuracy ofinterpolation with improved smoothing of statistical fluctuations may beachieved by superposing on the process described a component of thirdorder integration. This superposed integration is of identicalstep-by-step character, and is such that throughout the kth span, forall k for which the third order difference is defined, the summationcorresponds to the continuous integration expressed by ##EQU14## Theinvention may be extended on these lines to higher order of integrationif desired.

It has been observed that the derived interpolated function runssmoothly through the points of the original function, not necessarilypassing through any value which is not repeated a sufficient number oftimes. It is desired that the interpolated function should reliablystart and end on exact values that may be arranged by preceding thefirst point and following the final point by similar values so that theyare repeated a number of times at least equal to the number of recurrentaddition processes used. Thus if a function is to reliably start at andreturn to zero it can, for an interpolation using second orderdifferences, be preceded by two zeros and followed by two zeros.

Although the invention has been described in terms of absorption dataprocessed by a convolution method, it is applicable to any processingmethod requiring interpolation between a limited number of derived data.

What I claim is:
 1. Medical radiographic apparatus, for constructing arepresentation of the distribution of a characteristic of a body withrespect to penetrating radiation transmitted through a slice of a bodyof a patient including: radiographic examination means for providingdata signals representing the intensity of radiation transmitted throughthe material of the body in different directions in the slice; initialprocessor means co-operating with the said radiographic examinationmeans for converting the said data signals into a plurality of signalseach representing the line integral of absorption of the radiation bymaterial along a respective one of a plurality of beam paths in theslice, the paths being distributed in a plurality of sets ofsubstantially parallel paths; data store means for storing the saidsignals representing line integrals at locations corresponding to therespective paths; processor means for modifying the said signalsrepresenting line integrals in response to other signals representingline integrals along paths of the same set of parallel paths to formfurther signals suitable for combination to provide the saidrepresentation; differencing circuit means for forming differencesignals, for the further signals of each set of parallel paths,representing the differences of the further signals of the respectiveset extending to order m, where m is at least two; integrator means forproviding interpolated difference signals and for combining the saidinterpolated difference signals by a process of recurrent addition toprovide interpolated further signals, of the same form as the saidmodified line integrals but corresponding to paths interpolated betweenand parallel to the paths of the respective set; and matrix store meanshaving a plurality of locations associated with a plurality ofpredetermined points in a field notionally delineated in the slice andfor storing at each location the values of the further signals fororiginal or interpolated beam paths whose centrelines pass within apredetermined distance of the respective point.
 2. A medicalradiographic apparatus according to claim 1 in which the radiographicexamination means includes: support for the patient's body; a framemounted for rotation about an axis intersecting the said slice; a sourceof X-rays supported by said frame so as to irradiate the said slice;detector means adapted to receive radiation from said source afterpassage through the body; means for moving the frame relative to thesaid support about said axis so as to direct the radiation through theslice from a plurality of different directions to provide the said datasignals.